Ap. Markeyev, THE CRITICAL CASE OF 4TH-ORDER RESONANCE IN A HAMILTONIAN SYSTEM WITHONE DEGREE-OF-FREEDOM, Journal of applied mathematics and mechanics, 61(3), 1997, pp. 355-361
The motion of a time-periodic Hamiltonian system with one degree of fr
eedom in the neighbourhood of an equilibrium position is studied. It i
s assumed that the equilibrium is stable in the first approximation an
d that fourth-order resonance is present. The critical case is conside
red, when the system parameters are such that, in order to draw rigoro
us conclusions about the stability of the equilibrium, terms of order
higher than four in the series expansion of the Hamiltonian must be ta
ken into account. Sufficient conditions are derived for stability and
instability, and the bifurcations of periodic motions are investigated
in the neighbourhood of the equilibrium position when the system para
meters pass through values corresponding to the critical case. The res
ults are applied in the problem of the motion of a sphere in a uniform
gravity field when there are collisions with the surface of an ellipt
ic cylinder with a horizontal generator. (C) 1997 Elsevier Science Ltd
. All rights reserved.